I need to find the quality factor (ie. $Q = \frac{\text{energy stored}}{\text{energy dissipated per cycle}}$) associated to the Moon orbiting around the Earth. Looking around for this, lead me to the following table (from here):

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I think the Q-factor they list is NOT related to what I'm looking for (if so, the Moon would lose energy and crash into Earth quite quickly).

From what I have found out, the $k_{lm}$ numbers are Love numbers, and have something to do with the elasticity of a planet/moon in a particlar spherical-harmonic-type deformation. I am guessing that the Q-factor listed is the Q-factor related to mechanical oscillations of these deformations.

Could someone help explaining/clarifying/pointing me towards where I can find answer for:

1.- What exactly are the Love numbers ($k_{lm}$)?

2.- What are the Q factors that are being listed?

3.- How can I obtain the Q factor I am looking for?


1 Answer 1


You really should search Google first.

1) Check out Wikipedia. They are the ratios between the various spherical deformations and the deforming potentials (since astronomical bodies do not have perfectly spherically symmetric gravitational potentials, so you'll get higher order terms in a spherical expansion of the potential).

2) I am pretty sure they have the same definition that you give, where the energy stored is the one in a "tidal oscillator". Apparently, Springer gives a definition where the stored energy is the orbital energy, but this is as you've correctly noticed, impossible. Better understanding can be seen from the next google results: this paper, where section 2 gives a very good example of what Q is and this Caltech lecture, which explains the "tidal oscillator".

3) Given your question, I assume that what you are looking for is the ratio between the orbital energy of the Earth-Moon system and the energy lost every cycle. For this, you'll first want to find the energy lost every cycle. You should compute the deformation potentials created by the Moon and Earth (here is where Love numbers are used), then the work done to deform the Earth and Moon by each other's gravitational potentials, then divide by the relevant Q factor.


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